Action of Reduced Powers on the Homology of Products of Complex Projective Spaces and Products of Lens Spaces

The so-called ‘hit problem’ initiated by Peterson in [1] as an attempt at better understanding the $$E_2$$ -page of the Adams spectral sequence $$\operatorname{mod} 2$$ (that is the cohomology of the Steenrod algebra $$ {\mathcal{A}_2} $$ ) turned out to be very difficult. The hit problem is to determine a minimal generating set for the cohomology of products of infinite projective spaces $${\mathbb R} P^\infty$$ as a module over the Steenrod algebra $$ {\mathcal{A}_2} $$ at the prime 2. The dual problem is to determine the set of $$ {\mathcal{A}_2} $$ -annihilated elements in the homology of the same spaces. Anick showed that the set of $$ {\mathcal{A}_2} $$ -annihilated elements in the products of infinite projective spaces $${\mathbb R} P^\infty$$ forms a free associative algebra [6]. Ault and Singer proved that, for every $$k \ge 0$$ , the set of $$k$$ -partially $$ {\mathcal{A}_2} $$ -annihilated elements in homology of products of $${\mathbb R} P^\infty$$ (that is a set of elements that are annihilated by $$Sq^{2^i}$$ for all $$i \le k$$ ) also forms a free associative algebra. In this note, we investigate the dual problem at a prime $$p>2$$ . In this case, $${\mathbb R} P^\infty$$ should be replaced by $${\mathbb C} P^\infty$$ if one wants to ignore the action of the Bockstein operation $$\beta$$ or by the infinite $$p$$ -lens space $$L^\infty$$ to take $$\beta$$ into consideration. We prove that, for any $$k\ge 0$$ , a collection of elements in $${\mathbb Z}/p$$ -homology of products of $${\mathbb C} P^\infty$$ (or $$L^\infty$$ ) annihilated by all $$P^{p^i}$$ , $$i\le k$$ , forms a free algebra. The same holds for the collection of elements annihilated by $$\beta$$ and all $$P^{p^i}$$ , $$i\le k$$ . We also construct an explicit basis in the subspace $$\bar\Delta(0)_{m,*}\subset H_*(({\mathbb C} P^\infty)^{\wedge m},{\mathbb Z}/p)$$ , $$m=1, 2$$ , annihilated by $$P^1$$ . DOI 10.1134/S1061920825600230